Patterns, Squares and Circles and Algebraic Expressions
Haehl—Introductory Algebra
I had my students do this activity the third day in Introductory Algebra. The first day
students had done an activity about constants and variables that also had a similar pattern
recognition activity (attached at the end of this activity). In the first activity, we had a
follow-up discussion with the entire class and discussed which terms ended up as variable
patterns and which were constant. In this activity, each group had a slightly different
pattern. I gave each group a transparency to report about their particular activity and
graded each group with the rubric below.
The concepts/skills/notation that surfaced from the activity and follow-up discussion:
• Correct use of the words “variable” and “constant”
• Learning how to write “n × n” as n2
• Reinforcement of what a term is.
• What are “like terms?”
• What is a coefficient?
• How do you substitute a value into a formula?
• You can’t use the same variable to represent two different concepts—“n is the
card number, so it cannot represent the total circles and square on a card.”
This activity can be expanded to Intermediate or College Algebra by introducing function
notation, dependent and independent variable, graphing and list features of a graphing
calculator, talking about sequences, have more complicated patterns (decreasing patterns,
exponential) and by having students come up with at least 2 patterns established by the 3card sequence.
Student groups get the following grading sheet before they start the activity and I graded
the activity while students did the presentation.
Grading Rubric
Explanations are clear, writing is
easy to read, and correct
mathematical language is used.
Presenter communicates with the
class and makes eye contact.
Everyone participates and
contributes.
Total Points:
Excellent
3
Good
2
Satisfactory
1
3
2
1
3
2
1
Comment below or on the back about how your group worked together. (It is possible
that an individual who did not participate or was out of the room for a substantial part of
the discussion will receive reduced points.)
Patterns, Squares and Circles and Algebra Expressions
Haehl—Introductory Algebra
Group 1: Cards 1, 2, and 3 have the number of squares and circles as shown below.
Card 1
Card 2
Card 3
1. If the first three cards numbered 1, 2, and 3, have the following number of squares
and circles on the respective cards, fill in the table below to show how many circles
and squares would be on the next cards in the sequence.
Number of
Number of
Total Squares
Card Number
Squares
Circles
and Circles
1
2
3
4
5
6
2. Using the variable, n, to write a mathematical formula to show how many total circles
and squares would be on the nth card.
3. Show how to substitute into the formula above to determine how many circles and
squares would be on the 1000th card.
4. Fill in the information on the transparency provided and present your problem to the
class.
Patterns, Squares and Circles and Algebra Expressions
Haehl—Introductory Algebra
Group 2: Cards 1, 2, and 3 have the number of squares and circles as shown below.
Card 1
Card 2
Card 3
1. If the first three cards numbered 1, 2, and 3, have the following number of squares
and circles on the respective cards, fill in the table below to show how many circles
and squares would be on the next cards in the sequence.
Number of
Number of
Total Squares
Card Number
Squares
Circles
and Circles
1
2
3
4
5
6
2. Using the variable, n, to write a mathematical formula to show how many total circles
and squares would be on the nth card.
3. Show how to substitute into the formula above to determine how many circles and
squares would be on the 1000th card.
4. Fill in the information on the transparency provided and present your problem to the
class.
Patterns, Squares and Circles and Algebra Expressions
Haehl—Introductory Algebra
Group 3: Cards 1, 2, and 3 have the number of squares and circles as shown below.
Card 1
Card 2
Card 3
1. If the first three cards numbered 1, 2, and 3, have the following number of squares
and circles on the respective cards, fill in the table below to show how many circles
and squares would be on the next cards in the sequence.
Number of
Number of
Total Squares
Card Number
Squares
Circles
and Circles
1
2
3
4
5
6
2. Using the variable, n, to write a mathematical formula to show how many total circles
and squares would be on the nth card.
3. Show how to substitute into the formula above to determine how many circles and
squares would be on the 1000th card.
4. Fill in the information on the transparency provided and present your problem to the
class.
Patterns, Squares and Circles and Algebra Expressions
Haehl—Introductory Algebra
Group 4: Cards 1, 2, and 3 have the number of squares and circles as shown below.
Card 1
Card 2
Card 3
1. If the first three cards numbered 1, 2, and 3, have the following number of squares
and circles on the respective cards, fill in the table below to show how many circles
and squares would be on the next cards in the sequence.
Number of
Number of
Total Squares
Card Number
Squares
Circles
and Circles
1
2
3
4
5
6
2. Using the variable, n, to write a mathematical formula to show how many total circles
and squares would be on the nth card.
3. Show how to substitute into the formula above to determine how many circles and
squares would be on the 1000th card.
4. Fill in the information on the transparency provided and present your problem to the
class.
Patterns, Squares and Circles and Algebra Expressions
Haehl—Introductory Algebra
Group 5: Cards 1, 2, and 3 have the number of squares and circles as shown below.
Card 1
Card 2
Card 3
1. If the first three cards numbered 1, 2, and 3, have the following number of squares
and circles on the respective cards, fill in the table below to show how many circles
and squares would be on the next cards in the sequence.
Number of
Number of
Total Squares
Card Number
Squares
Circles
and Circles
1
2
3
4
5
6
2. Using the variable, n, to write a mathematical formula to show how many total circles
and squares would be on the nth card.
3. Show how to substitute into the formula above to determine how many circles and
squares would be on the 1000th card.
4. Fill in the information on the transparency provided and present your problem to the
class.
Patterns, Squares and Circles and Algebra Expressions
Haehl—Introductory Algebra
Group 6: Cards 1, 2, and 3 have the number of squares and circles as shown below.
Card 1
Card 2
Card 3
1. If the first three cards numbered 1, 2, and 3, have the following number of squares
and circles on the respective cards, fill in the table below to show how many circles
and squares would be on the next cards in the sequence.
Number of
Number of
Total Squares
Card Number
Squares
Circles
and Circles
1
2
3
4
5
6
2. Using the variable, n, to write a mathematical formula to show how many total circles
and squares would be on the nth card.
3. Show how to substitute into the formula above to determine how many circles and
squares would be on the 1000th card.
4. Fill in the information on the transparency provided and present your problem to the
class.
Patterns, Squares and Circles and Algebra Expressions
Haehl—Introductory Algebra
Group 7: Cards 1, 2, and 3 have the number of squares and circles as shown below.
Card 1
Card 2
Card 3
1. If the first three cards numbered 1, 2, and 3, have the following number of squares
and circles on the respective cards, fill in the table below to show how many circles
and squares would be on the next cards in the sequence.
Number of
Number of
Total Squares
Card Number
Squares
Circles
and Circles
1
2
3
4
5
6
2. Using the variable, n, to write a mathematical formula to show how many total circles
and squares would be on the nth card.
3. Show how to substitute into the formula above to determine how many circles and
squares would be on the 1000th card.
4. Fill in the information on the transparency provided and present your problem to the
class.
Patterns, Squares and Circles and Algebra Expressions
Haehl—Introductory Algebra
Group 8: Cards 1, 2, and 3 have the number of squares and circles as shown below.
Card 1
Card 2
Card 3
1. If the first three cards numbered 1, 2, and 3, have the following number of squares
and circles on the respective cards, fill in the table below to show how many circles
and squares would be on the next cards in the sequence.
Number of
Number of
Total Squares
Card Number
Squares
Circles
and Circles
1
2
3
4
5
6
2. Using the variable, n, to write a mathematical formula to show how many total circles
and squares would be on the nth card.
3. Show how to substitute into the formula above to determine how many circles and
squares would be on the 1000th card.
4. Fill in the information on the transparency provided and present your problem to the
class.
What is a Variable, What is a Constant?
Martha Haehl
Introductory Algebra
1. What does the words “variable” mean to you outside of a math class? Describe a
way the word, variable, might be used in a newscast or newspaper, a magazine, or
in everyday conversation.
2. What does the words “constant” mean to you outside of a math class? Describe a
way the word, constant, might be used in a newscast or newspaper, a magazine, or
in everyday conversation.
3. Describe what the word, variable, means to you in mathematics.
4. Describe what the word, constant, means to you in mathematics.
Pattern Recognition, Constants and Variables
Martha Haehl, Introductory Algebra
Scenario: Three creators—Jebula, Maliba, and Noble—of planet Outthere decided to
spiff up the globe by laying 3 colors of tile in various patterns on the surface. Each
creator chose her favorite color and devised her own scheme for laying the tiles. On
the 1st three days the creators laid tiles as shown in the table below. (The tiles were
slightly curved so that they fit on the curved surface of Outthere.)
Day 1
Day 2
Day 3
Jebula (Red)
1 tile
1 tile
1 tile
Maliba (Purple)
1 tile
2 tiles
3 tiles
Noble (Green)
1 tile
2×2 square of tiles
3×3 square of tiles
1. Following a pattern implied in the table, how many tiles of each color were laid
on the 3rd, 4th, 5th, 6th, 7th, 8th, 9th, and10th days, 19th day, 108th day, 1095th
day? How many total tiles were laid for each day?
Red
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Day 19
Day 108
Day 1095
Purple
Green
Total Tiles
2. In words, explain to someone how he or she would calculate how many tiles of
each color and the total number of tiles that would be laid in one day—without
knowing in advance which day it is (the 1st, 2nd, 99th, for example).
3. Use your description from Question 2 to write a formula to calculate the total
number of tiles that would be laid on the nth day. Identify the constant and
variable terms in your formula.
Total number of tiles laid on nth day: ___________________________
(formula)
4. For each creator, describe whether you would categorize the number of tiles laid
each day as "constant" or "variable." Explain how you made the determination.
Constant?
Variable?
Jebula (Red)
Maliba (Purple)
Noble (Green)
Explanation